NBER WORKING PAPER SERIES

REDISTRIBUTION BY INSURANCE MARKET REGULATION:ANALYZING A BAN ON GENDER-BASED RETIREMENT ANNUITIES

Amy FinkelsteinJames Poterba

Casey Rothschild

Working Paper 12205http://www.nber.org/papers/w12205

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138April 2006

We are grateful to Jeff Brown, Pierre-Andre Chiappori, Keith Crocker, Peter Diamond, Liran Einav, MikhailGolosov, Robert Gibbons, Kenneth Judd, Whitney Newey, Bernard Salanie, and participants at the NBERInsurance Meeting, the Stanford Institute for Theoretical Economics, and the Econometric Society AnnualMeeting for helpful discussions, to Luke Joyner and Nelson Elhage for research assistance, and to theNational Institute of Aging and the National Science Foundation (Poterba and Rothschild) for researchsupport. The views expressed herein are those of the author(s) and do not necessarily reflect the views of theNational Bureau of Economic Research.

©2006 by Amy Finkelstein, James Poterba and Casey Rothschild. All rights reserved. Short sections of text,not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including© notice, is given to the source.

Redistribution by Insurance Market Regulation: Analyzing a Ban on Gender-Based RetirementAnnuitiesAmy Finkelstein, James Poterba and Casey RothschildNBER Working Paper No. 12205April 2006JEL No. D82, H55, L51

ABSTRACT

This paper shows how models of insurance markets with asymmetric information can be calibratedand solved to yield quantitative estimates of the consequences of government regulation. Weestimate the impact of restricting gender-based pricing in the United Kingdom retirement annuitymarket, a market in which individuals are required to annuitize tax-preferred retirement savings butare allowed considerable choice over the annuity contract they purchase. After calibrating a lifecycleutility model and estimating a model of annuitant mortality that allows for unobserved heterogeneity,we solve for the range of equilibrium contract structures with and without gender-based pricing.Eliminating gender-based pricing is generally thought to redistribute resources from men to women,since women have longer life expectancies. We find that allowing insurers to offer a menu ofcontracts may reduce the amount of redistribution from men to women associated with gender-blindpricing requirements to half the level that would occur if insurers were required to sell a singlepre-specified policy. The latter "one policy" scenario corresponds loosely to settings in whichgovernments provide compulsory annuities as part of their Social Security program. Our findingssuggest that recognizing the endogenous structure of insurance contracts is important for analyzingthe economic effects of insurance market regulations. More generally, our results suggest thattheoretical models of insurance market equilibrium can be used for quantitative policy analysis, notsimply to derive qualitative findings.

Amy FinkelsteinDepartment of EconomicsMIT E52 262F50 Memorial DriveCambridge MA 02142and NBERafink@mit.edu

James M. PoterbaDepartment of EconomicsMIT, E52-35050 Memorial DriveCambridge, MA 02142-1347and NBERpoterba@mit.edu

Casey RothschildMIT Department of EconomicsOffice Number E51-390E52, Room 391Cambridge MA 02142-1347 caseyr@mit.edu

1

Restrictions on the use of characteristics such as race or gender in pricing are ubiquitous in private

insurance markets. These restrictions are likely to become even more important as the advent of genetic

tests enriches the information set that insurers might use to price life and health insurance policies.

Several theoretical studies, including Hoy (1982) and Crocker and Snow (1986), have analyzed this form

of regulation and shown qualitatively that they have unavoidable negative efficiency consequences.

Empirical work such as Buchmueller and DiNardo (2002) and Simon (forthcoming) has confirmed the

existence of such efficiency costs by documenting declines in insurance coverage when characteristic-

based pricing is banned in health insurance markets. However, there have been few if any attempts to

develop quantitative estimates of the efficiency costs or the distributional impacts of restrictions on

characteristic-based pricing. One of the few studies in this vein is Blackmon and Zeckhauser’s (1991)

analysis of automobile insurance regulation. It frames questions similar to the ones we study but does not

analyze how the structure of insurance contracts may respond to regulatory restrictions or how this affects

distributional or efficiency effects.

In this paper, we take a first step toward developing quantitative estimates of the effects of

endogenous contract responses to insurance market regulation. We extend existing theoretical models and

adapt them to provide quantitative estimates of both the efficiency and redistributive effects of a unisex

pricing requirement for pension annuities. Restrictions on characteristic-based pricing are usually thought

to transfer resources from individuals in lower-risk categories to those with greater risks. Women are

longer-lived than men, so unisex pricing restrictions in the pension marketplace redistribute from men to

women. Some might argue for such policies on redistributive grounds, since elderly women have higher

poverty rates than elderly men. Viewed from the ex-interim perspective once individual characteristics

are known, the transfers from men to women generate redistribution akin to the redistribution associated

with uniform pricing regulations in industries such as telephone and electricity distribution, where

individuals have different costs of service. Posner (1971) labeled such redistribution “taxation by

regulation.” Alternatively, from an ex-ante perspective before individual characteristics are known, the

2

redistribution may be viewed as a form of insurance against drawing a high-cost characteristic, in this

case being female, as in Hirshleifer (1971).

In addition to providing a tractable setting for illustrating our techniques, the pension annuity market

is an interesting setting in its own right because of its size, its importance for retiree welfare, and the

salience of unisex pricing regulations in this market. Private annuity arrangements, typically the payouts

from defined benefit pension plans, represent an important source of retirement income for many elderly

households. Employers in the United States were once free to offer different pension annuity payouts to

men and women, but litigation in the 1970s and early 1980s eliminated this practice. The European Union

is currently debating regulatory reforms that may eliminate gender-based pricing in insurance markets,

including pension annuity markets. Our analysis may also have broader implications for the design and

regulation of annuitized payout structures associated with defined contribution Social Security systems.

We are not aware of any previous attempts to calibrate and solve stylized theoretical models of

insurance market equilibria. Doing so requires adapting these models to account for a number of features

that are observed in actual insurance markets. One that has quantitatively important implications is our

relaxation of the assumption that individuals have no recourse to an informal, if inefficient, substitute for

insurance. Our analysis recognizes that individuals may save against the contingency of a long life, and

that insurance companies many not observe savings by their policyholders. If we do not allow for

unobservable savings, the informational asymmetries created by a ban on gender categorization may have

neither efficiency nor distributional effects.

We focus on the retirement annuity market in the United Kingdom, where we have obtained a rich

micro-data set that facilitates our calibration. A critical feature of this market is that workers who have

accumulated tax-preferred retirement savings must purchase an annuity. They cannot choose whether or

not to participate in the annuity market, which eliminates one margin on which unisex pricing regulations

could potentially affect individual behavior. Participants do have substantial flexibility with regard to

contract choice. Empirical evidence, such as that presented in Finkelstein and Poterba (2004), suggests

that this choice is affected by private information about risk type.

3

Our main finding is that recognizing the endogenous response in the structure of insurance contracts

when regulations change may reduce by as much as fifty percent the amount of redistribution away from

men and toward women that would be associated with a ban on gender-based annuity pricing in a fully

compulsory annuity market with no scope for this response; this latter setting in which insurers are

required to sell a single pre-specified policy loosely corresponds to settings in which governments provide

compulsory annuities as part of their Social Security program. Our findings highlight the importance of

recognizing the endogenous structure of insurance contracts when analyzing the economic effects of

insurance market regulation, and they indicate that theoretical models of insurance market equilibrium

can be adapted to offer quantitative predictions on regulatory issues. Even accounting for the endogenous

contract response, however, we find that a ban on gender-based pricing in the U.K. retirement annuity

market would have substantial distributional consequences, in most cases redistributing at least three

percent of retirement wealth from men to women. We also estimate that the efficiency costs associated

with this redistribution would be very small. However, since individuals do not have a choice of whether

or not to participate in this market, our estimates of the efficiency costs of unisex pricing restrictions are

likely to substantially underestimate the cost of such restrictions in voluntary annuity markets.

Our analysis is divided into six sections. The first briefly reviews the qualitative impact of uniform

pricing requirements in insurance markets with asymmetric information. Based on the assumption that

annuity markets operate in a constrained-efficient manner, section two develops a model of the range of

possible contracts offered and purchased in equilibrium. It also describes results concerning equilibrium

contract structure and our algorithm for solving for these contracts. It is supplemented by a technical

appendix. In the third section we calibrate the model and describe our estimates of a two-type mixture

model for mortality rates. Section four describes the measures that we use for evaluating the efficiency

and distributional effects of policy interventions in insurance markets. The fifth section presents our

quantitative results. We describe the range of possible distributional and efficiency effects of restrictions

on gender based pricing under different assumptions concerning the constraints on consumers and

4

producers. A brief conclusion discusses how our results bear on a number of ongoing policy debates and

describes possible generalizations of our approach to other insurance markets.

1. A Framework for Analyzing Regulation in Insurance Markets

This section reviews the qualitative efficiency and distributional effects of a ban on categorization in

a standard two-state, two-type model of competitive insurance markets with asymmetric information.

This framework considers two distinct types of individuals who are indistinguishable to an insurance

company but who face different risks of a loss. Individuals can insure themselves against loss by

purchasing a single insurance contract from firms in a competitive market.

1.1 Qualitative Analysis of Banning Categorization in the “Perfect Categorization” Case

There is little consensus concerning the proper equilibrium concept for insurance markets with

asymmetric information, as Hellwig (1987) explains. We therefore follow the approach taken by Crocker

and Snow (1986) in their analysis of the efficiency impacts of bans on categorization and focus on

constrained efficient outcomes. In focusing on these outcomes, we implicitly assume that the private

market achieves efficient outcomes, within the scope of their ability to do so, without explicitly modeling

equilibrium behavior. We note, however, that the so-called Miyazaki (1977)-Wilson (1977)-Spence

(1978) (hereafter MWS) equilibrium provides an example of a model of equilibrium behavior that results

in a constrained efficient outcome. We will describe this MWS outcome in more detail after

characterizing the entire efficient frontier, as it will play an important role in our analysis.

To characterize the frontier, denote the high risk and low risk types by H and L, respectively. Let

)(AV i denote the indirect utility achieved by type i when she has purchased insurance contract A, and let

)(AiΠ denote the expected profits a firm earns by selling contract A to type i. With this notation, points

on the Pareto frontier solve the following program, where λ is the proportion of H types:

5

(1)

,0)()()1()()()(

)()()(

)()()(

)(max,

≥Π+Π−

≥

≥

≥

HHLL

HHH

HLLLL

LHHHH

LL

AA

AABCVAVMU

AVAVIC

AVAVICtosubject

AVHL

λλ

where (ICi) is the incentive compatibility constraint stating that i types must be willing to choose the

contract designed for them, (BC) is a budget constraint that requires that on average policies break even,

and (MU) is a minimum utility constraint for the H types.

Crocker and Snow (1985) characterize this constrained Pareto frontier in the standard two period

(one-accident) setting by varying the Lagrange multiplier on constraint (MU) in (1). In Figure 1, we

characterize the frontier in the same two-period setting by varying the value of HV . Insurance contracts

can be written as state-contingent consumption vectors ),( 10 aaA = , where the subscript 0 refers to the

“no accident” state and the subscript 1 refers to the “accident” state. Insurance providers supply these

consumption promises A in exchange for a buyer’s state-contingent endowment wealth vector

),( 00 l−= wwW . H types have a higher probability of experiencing state 1 and the types are otherwise

identical expected utility maximizers with a strictly concave utility function.

For low values of HV , (MU) may be slack. For example, if { }

)(max0)(:

AVV H

AA

HH =Π

= so that (MU)

says that H types have to be at least as well off as they would be with their full insurance actuarially fair

consumption point, then (MU) will be slack precisely when the Rothschild and Stiglitz (1976) equilibrium

either fails to exist or exists but fails to be constrained efficient. Such a situation is depicted at point M in

Figure 1. At point M, L types consume the constrained efficient allocation that is best for them; this

corresponds to the MWS equilibrium. Figure 1 shows that even this best-for-L allocation can involve

positive cross subsidies from the Ltypes to the H types.

6

The dark curve connecting points M and F in Figure 1 depicts a portion of the locus of the L type

consumption points that correspond to constrained Pareto optimal outcomes. The point labeled F is the

unique “pooling” outcome on the frontier – i.e., the unique constrained efficient outcome with HL AA = .

It is on the 45-degree line and therefore provides full insurance. Point F involves substantially larger

cross subsidies from L types to H types than does M. There are additional constrained efficient outcomes

not depicted in Figure 1 which involve even larger cross subsidies from L type to H types than those at

point F. Such outcomes involve the L types being fully insured and the H types being overinsured, which

Crocker and Snow (1985) note is a feature absent from standard models of equilibrium in insurance

markets. As a result, we do not consider this portion of the frontier. The set of outcomes we consider is

thus captured in the region of the frontier bounded by F and M; we do not try to select any particular

constrained efficient outcome from this set.

Because (1) permits – and, as in the case of Figure 1, may even require – the market to implement a

contract pair involving cross subsidies across types, bans in characteristic-based pricing can have both

distributional and efficiency consequences. This is illustrated in Figure 2, which depicts a constrained

efficient pair of contracts. When type is observable and can be used in pricing, the competitive

equilibrium will provide each type with her actuarially fair full insurance contract. In Figure 2, *HA and

*LA depict the full insurance actuarially fair contracts that we assume emerge when type is observable

and can be contracted upon. Consumption for each type is independent of the realized state of nature.

When type-based pricing is banned, our assumption is that the market implements a pair of contracts,

labeled HA and LA , which is constrained efficient given the informational restrictions of the ban. Note

that as depicted this contract pair involves positive cross subsidies between types. As a result, H types are

better off when categorization is banned, and L types are worse off. This illustrates how a ban on

categorical-based pricing may have distributional consequences. The ban is efficiency reducing in this

example as well. Since type is, in fact, observable, it is in principle possible to make L types as well off

as with LA via contract LA′ , which is also actuarially cheaper to provide to the L types.

7

1.2 Residual Private Information

The foregoing discussion assumes that type is observable. A ban on characteristic-based pricing

therefore moves the economy from perfect information to imperfect information. In practice, information

such as gender or the outcome of a genetic test may be related to risk type, but even conditional on this

information, insurers are unlikely to be able to completely determine the risk of potential policy buyers.

The relevant comparison is therefore between imperfect information and more imperfect information.

Our study builds on previous analyses of bans on characteristic-based pricing, such as Hoy (1982)

and Crocker and Snow (1986), which use the most parsimonious model that can capture the presence of

residual uncertainty. There are still two risk types, but risk type is not directly observable. Instead,

insurers only observe a signal that is correlated with risk type. There are two possible signals, X and Y.

We henceforth refer to individuals as falling in category X or category Y. A fraction kλ of category k

individuals are high risk types, with .10 <<< YX λλ Thus, category Y is the high-risk category, but

there are still low-risk types within that category. We denote by θ the fraction of category Y individuals

in the population.

For our analysis, we continue to assume that markets will operate in a constrained efficient manner

given the information which is both available and legal for use in pricing. When characteristic-based

pricing is permitted, we further assume that the market will not implement contracts involving cross-

subsidies across observable categories, just as we did in Figure 2 by assuming that the contracts *HA and

*LA emerge when type-based pricing was allowed. A ban on categorical pricing in this imperfect-

information setting will have the same qualitative effects as it does in the perfect information setting

described above.

2. Modeling Restrictions on Gender-Based Pricing in the U.K. Pension Annuity Market

The preceding discussion illustrates the qualitative impact of a ban on categorization on efficiency

and redistribution. To develop quantitative estimates, we consider a particular ban on categorization in a

8

particular market, namely the imposition of unisex pricing requirements in the U.K. annuity market.

Individuals in the United Kingdom with defined contribution private pension plans that have benefited

from tax deferral on investment income—the analogues of IRAs and 401(k)’s in the United States—face

compulsory annuitization requirements for a substantial share of the balance accumulated by retirement.

In 1998, data from the Association of British Insurers (1999) suggest that annual annuity payments in this

market totaled £5.4 billion.

Although annuitization is compulsory, annuitants in the U.K. retirement annuity market have some

scope for self-selection across contract choice. Finkelstein and Poterba (2004, 2006) find that such self-

selection appears to reflect private information about mortality risk. Note that, from the perspective of an

insurance company, high-risk annuitants are those who are likely to live longer than the characteristics

used in pricing, such as age and gender, would suggest. There are currently no regulations in the U.K.

annuity market limiting the characteristics used in pricing annuities. In practice, annuities are priced

almost exclusively on age at purchase and gender. Several small firms entered the annuity market after

the end of our sample with discounted annuities for heavy smokers, but those products were not available

during the period that we study.

While the two-state model discussed above suffices for the understanding the qualitative impacts of

interventions that ban categorical pricing, it is too stylized to plausibly measure the quantitative impact of

regulatory interventions. Since an individual can live for many years after the purchase of their annuity,

we extend the analysis to 35 periods. Boadway and Townley (1988) is the only other contract theoretic

model we have found that includes more than three periods in an analysis of an annuity market with

asymmetric information, but the contracts under consideration have a particular and restrictive form that

we relax. This extension to many periods is essential for a plausible calibration.

Our baseline model also allows for unobservable savings. Eichenbaum and Peled (1987), Brunner and

Pech (2005), and others note that allowing annuitants to engage in unobservable saving limits the ability

of insurers to screen different types of observationally equivalent annuity buyers. In our context, we show

that when insurance companies can observe savings, the informational asymmetries created by a ban on

9

gender categorization can have neither efficiency nor distributional consequences. The process of

deriving and solving the model, which we discuss below, provides insight into why accounting for

unobservable savings is critical for any plausible calibration. It also demonstrates why this extension

makes the model substantially more difficult to solve. We show that it is nevertheless possible to solve for

the contracts on the constrained Pareto frontier, and we sketch our computational algorithm.

2.1 Defining Annuity Market Outcomes

Our model applies to any number of periods Nt ,,0 L= , where we interpret t as the number of years

after retirement, which we take to be at age R=65. In practice, we take N=35, thereby assuming

individuals do not live past age 100. To capture the compulsory purchase requirement, we assume that

individuals must use their retirement wealth W to purchase an annuity. They exponentially discount the

future at rate r+= 11δ per year, where r is the interest rate, and by their (cumulative) probability tS of

living to a given age R+t. The two risk types, H and L, differ only in their survival probabilities. There is

a continuum of individuals, with a fraction λ of H types. We assume Lt

Lt

Ht

Ht

SS

SS 11 ++ > for each t; in other

words, the ratio of the cumulative survival probabilities of the two types must be monotone in age. This

is satisfied if the higher longevity type has a lower mortality hazard at every age.

The direct utility of a consumption stream ( )Ncc ,,0 L=Γ for type σ is given by:

(3) ∑∑=

−

= −==

N

t

tt

tN

ttt

tN

cScuSccU

0

1

00 1

)(),,(γ

δδγ

σσσ L ,

where γ is the risk-aversion parameter. Annuity streams, which are denoted by A, specify a life-

contingent payment ta in each of the N +1 periods. In our baseline model, we impose no structure on the

annuity payments ta ; we later restrict the time profile of possible annuity payments.

Individual savings earn an interest rate r. Individuals have no bequest motive, and they cannot borrow

against their annuity. This means that individuals with an annuity stream A can obtain any consumption

10

stream that satisfies ⎭⎬⎫

⎩⎨⎧

∀≤Γ≡∈Γ ∑∑ tacAFt

ss

t

ss

00)( δδ . This induces indirect utility functions and

type-specific actuarial cost functions

(4) )(max)()(

Γ=∈Γ

σσ UAVAF

,

and

(5) ∑≡N

ttn aSAC

0

)( σσ δ .

Because individuals discount the future at the rate of interest, “full insurance” annuities have level real

payouts. Let )(XV σ denote the utility that type σ gets by consuming the full insurance annuity A

with XAC =)(σ . Let λA denote the pooled-fair full insurance annuity – i.e., the full insurance annuity

satisfying WACAC LH =−+ )()1()( λλ λλ . In a constrained efficient market, the two risk types

purchase a pair of annuities HA and LA that solve:

(6)

WACACBCVAVMU

AVAVIC

AVAVICtosubject

AV

HHLL

HHH

HLLLL

LHHHH

LL

AA HL

≤+−

≥

≥

≥

)()()1()()()(

)()()(

)()()(

)(max,

λλ

for some HV . We further assume that )()( λAVVWV HHH ≤≤ , so that H types are at least as well

off as they would be if they revealed their type, and are no better off than they would be under a pooled-

fair full insurance outcome. This range corresponds with the portion of the efficient frontier in Figure 1.

Solving (6) is non-trivial: it involves solving for the N +1 year-specific annuity payments for each of the

two types. Furthermore, the functions )(AV σ are themselves implicitly defined via (4), which is an

optimization problem over N +1 variables. Nevertheless, (6) is computationally tractable.

11

Several factors help us solve (6). First, the assumption that )( λAVV HH ≤ implies that the L type

incentive compatibility constraint will be slack at the solution. We therefore drop this constraint while we

are solving (6), and later verify that it is indeed satisfied. Likewise, the budget constraint (BC) trivially

binds at the optimum. Second, once the type-L (IC) constraint is dropped, it is easy to see that HA will be

a full insurance annuity. Any allocation with an HA that does not offer full insurance can improved upon

by replacing HA with the full insurance bundle HA~ for which )()~( HHHH AVAV = , as this replacement

affects (6) (sans (ICL)) only by making (BC) slack. Since HA is a full insurance annuity, we can

parameterize it by )( LL ACWT −≡ , the size of the cross-subsidy from L types to H types expressed in

per L type terms. For a given T, ( )TWVAV HHHλλ−+= 1)( , which means that the solution to (6) must

have TT ≥ , where T solves ( )TWVV HHλλ−+= 1 . This permits us to write (6) in the simpler form:

(7)

TWACBCTTMU

TWVAVCItosubject

AV

LL

HLH

LL

TAL

−≤

≥

+≤′ −

)()'()'(

)()()(

)(max

1

,

λλ

In practice, we solve this program for a given T and then perform a search over different values of T to

find the optimum. In discussing (7), we therefore treat T as given.

Third, we observe that neither type chooses to save at an efficient contract pair. This is obvious for H

types since HA is a full insurance annuity. The L types have no incentive to save in a constrained

efficient market because saving is an inherently inefficient mechanism for transferring income forward in

time when there is no bequest motive. It is more efficient to use life-contingent payments so that

resources are not “wasted” at death. If an L type receives an annuity LA that induces her to save at some

age, then her consumption stream, say LA~ , would differ from the annuity stream. That same

consumption stream could be achieved directly via an annuity at a lower actuarial cost to the annuity

12

provider. There is therefore some surplus to be created by reducing the annuity’s payouts in its early

years and raising its payouts in later years. Insurers in an efficient market will take advantage of such

opportunities to repackage the timing of cash flows until the surplus is eliminated and L types no longer

wish to save from the annuity. Formally, consider replacing LA with LA~ in (7). L types would be

exactly as well off as before, but when LL AA ~≠ the budget constraint would be made strictly looser.

Furthermore, the incentive compatibility constraint will be no tighter, and possibly strictly looser, as a

result of the replacement. Therefore, LA can only solve (7) when LL AA ~= .

The observation that neither type chooses to save means that, in equilibrium, )()( LLLL AUAV = and

)()( HHHH AUAV = , so both can be computed directly instead of by solving the non-trivial (4). The

only part of (7) that is difficult to compute is )( LH AV , the utility that H types get if they deviate to

purchasing the L type annuity and saving optimally. The structure of (7) in fact allows us to evaluate

)( LH AV in solving for equilibrium without explicitly solving (4). In particular, with the parametric forms

we assume on the survival probabilities and preferences, );(~)( *nAVAV LHLH = at any solution to (7)

for some n*, where

(8) ∑=

=N

t

Ht

Ht

tL

H cuSnAV0

* )~();(~ δ

and where

(9)

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≥

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

<

=

∑

∑

=

=*

*

*

1

*

*

1

*

~

ntif

SS

aSS

ntifa

c

Nnn H

n

Hnn

Nnn

Ln

nHn

Ht

Lt

Ht

γ

γ

δ

δ

Equations (8) and (9) describe the utility achieved by an H type with an annuity stream LA when she

consumes the payments before period n*, and thereafter follows the consumption pattern she would

13

follow if the remaining annuity stream ( )LN

Ln

aa ,,* L were a bond against which she could save and

borrow at the constant rate r. Hence, saying that );(~)( *nAVAV LHLH = for some n* at a solution to (7)

is tantamount to saying that that the optimal consumption pattern of H types who deviate and buy annuity

stream LA is of this form. Note that for their utility to be given by a consumption pattern of this form, the

stream LA must be such that this consumption pattern of deviating H types does not involve borrowing.

The formal proof that annuity stream LA has the property that deviating H types will optimally

consume in accord with (9) is shown in the appendix. The intuition is relatively straightforward,

however, and it offers insights into the critical importance of saving in determining the optimum annuity

streams. Suppose that annuitants could not save. Then we could find the solution to (7) by simply

replacing )( LH AV with )( LH AU . This modified program could be solved using first order conditions.

To illustrate such a solution, Figure 3 plots the annuity streams LA and HA for a special case of the

general problem, corresponding to the 0=T extreme (i.e. the MWS equilibrium) and to the male

population in the baseline parameterization of our model, as developed below. The special case also

assumes 3=γ and r = .03. Figure 3 shows that HA is a full insurance annuity, and LA is an annuity

which is almost a full insurance annuity with significantly higher annuity payments. The payments

provided by LA decline with time, but this decline is only significant at late ages – indeed, the decrease is

negligible until age 97. The payments fall off sharply thereafter, but the AL annuity payment only falls

below the AH annuity payment at age 100 – the oldest age considered. Between ages 99 and 100,

however, the payment falls off so sharply that the incentive compatibility constraint is nevertheless

satisfied. Qualitatively similar plots would hold for less extreme values of .T

The reason the annuity stream LA falls off so steeply and at such an advanced age is because this is

when HL

SS is smallest. Low annuity payments translate directly into low consumption when savings is

impossible; this hurts H types much more than L types at old ages, since H types are relatively much

14

more likely to still be alive. In other words, the best way from the perspective of L types to satisfy

incentive compatibility for H types involves providing a downward tilt at extreme old ages, when the

relative probability of L types being alive, compared to H types, is lowest.

When savings is possible, such a steep drop-off is far less useful as a self-selection device because it

can always be undone – albeit inefficiently – by saving. Indeed, Figure 3 also shows the optimal

consumption pattern Htc~ and bond-wealth holding of H types who receive annuity LA but who can also

save. These H types optimally choose to consume the annuity payments until age 96, after which they use

their savings to smooth out the sharp drop-off in the annuity stream. Because such saving reduces the

power of downward-sloping payout schedules as a selection device, when savings is possible, the

extremely sharp fall-off of payments LA will no longer be optimal. However, the incentive for positive

saving by deviating H types will still be as in (9).

2.2 Optimal Structure of Contracts

A central contribution of our modeling is finding the optimal structure of annuity contracts when

annuitants can save. This involves solving (7). We cannot offer general analytic solutions, so our

findings necessarily require assumptions about the underlying functional forms of the utility function, the

mortality rates, and other parameters. Using the same baseline parameters that we used in Figure 3, and

the same assumption that 0=T , Figure 4 plots the solution to (7) and shows the actuarially fair full

insurance annuities for both H type and L type individuals, as well as the optimal consumption stream of

an H type who deviates and purchases annuity LA . Again, qualitatively similar graphs would obtain for

other values of .T

Several features of Figure 4 are worthy of note. First, the solution involves substantial cross-

subsidies. This is clear from the comparison of the level of the H type fair level annuity and the H type

optimum annuity HA , as HA offers strictly higher payouts. Second, while LA provides a downward

sloping annuity stream, it declines much more gradually than the annuity stream shown in Figure 3,

which corresponded to the case in which annuitants could not save. Third, comparison of the optimal

15

consumption stream of an H type deviating to LA reveals that the deviating H type who purchases LA

will immediately begin to save. In the notation above, this means 0* =n in (8) and (9).

Comparison of Figures 3 and 4 shows the important effect of allowing for unobservable saving on the

structure of the optimal annuity streams. Though it is more difficult to find the optimal annuities with

unobservable saving than without, the evident realism that allowing for such saving provides leads us to

choose this as our benchmark case. Indeed, the results in Figure 3 suggest that if unobservable saving is

not possible, asymmetric information is essentially irrelevant because the optimal annuity streams are

virtually identical to the annuity streams that would obtain with symmetric information. The findings

more generally suggest caution in using applied contract theoretic models for quantitative purposes when

there are inefficient and unobservable behaviors the insured can undertake as a substitute for formal

insurance.

2.3 Discussion of Key Assumptions

The importance of unobservable savings highlights one of several extensions we have made to the

standard stylized model of insurance markets with asymmetric information. These extensions provide a

more realistic framework for analyzing the impact of a ban on gender-based pricing. Nonetheless, the

model that we develop in (6) and (7), and then solve, makes a number of assumptions for tractability.

Some – such as the use of constant relative risk aversion utility or the assumption that individuals

discount the future at the rate of interest – are standard. It is worth, however, briefly commenting on

several that are more specific to this application.

First, we have not incorporated bequest motives into our model. The importance of bequests in

explaining saving behavior has been widely debated, for example by Kotlikoff and Summers (1981),

Hurd (1987, 1989), Bernheim (1991), and Brown (2001), but no consensus has emerged. Conceptually,

the presence of bequest motives can easily be incorporated into our framework. We would simply add

utility from consumption in states when the consumer is dead. Since our solution algorithm relies heavily

on the shape of preferences, however, this extension can pose practical issues of computational

16

tractability. In part for this reason, we have addressed the analytically more convenient setting without

bequests, while recognizing that this limits the applicability of our findings if actual consumption

decisions are substantially affected by bequest motives.

Second, we have followed previous theoretical models, notably Hoy (1982) and Crocker and Snow

(1986), in modeling mortality heterogeneity via two risk types. The computational challenge of finding

optimal contracts is much more difficult in a many-type setting, although similar solution algorithms to

the ones we developed here would, in principle, also apply. We show below that our data cannot reject

this parsimonious model in favor of one which allows the underlying types to differ by gender.

Finally, we emphasize more generally that while our model incorporates some important features of

the U.K. annuity market, it does not capture many others. For example, we focus on single life annuities,

and we ignore individuals’ option to purchase limited term guarantees of their contracts. We also ignore

the presence of wealth outside the retirement accounts. We abstract from the possible presence of risks

other than longevity risk, such as liquidity risks or health shocks; Crocker and Snow (2005) discuss how

the existence of such “background risks” can affect the insurance market equilibrium. Finally, our model

does not allow for the possibility of individuals learning over time about their risk type; Polborn et al.

(2004) show that allowing for such dynamic considerations in a model in which individuals have

flexibility in the timing of their insurance purchases can have important qualitative effects for the impact

of restrictions on characteristic-based pricing. In part because of these and other abstractions, the optimal

annuity contracts we compute do not match the actual contracts observed in the data; we discuss this in

more detail below.

3. Model Calibration

Calibrating our model to yield quantitative estimates of the efficiency and distributional consequences

of mandating unisex prices requires the constant relative risk aversion parameter γ; the real interest rate r;

the fraction of high risk individuals among men ( Mλ ) and among women ( Fλ ); the fraction θ of women

17

in the relevant population; and the survival curves for each risk type ) and ( LH SS . We present results

for risk aversion coefficients of 1, 3 and 5. We assume the interest rate r is equal to 0.03 and set the

discount rate r+= 11δ . We set 5.0=θ in our baseline case, but we also report results for other values.

We jointly estimate the remainder of the parameters using micro-data on a sample of compulsory

annuitants who bought annuities from a large U.K. life insurance company between 1981 and 1998. We

have information on their survival experience through the end of 1998. These data, which are described in

more detail in Finkelstein and Poterba (2004), appear to be reasonably representative of the U.K. annuity

market. We restrict our attention to annuities that insure a single life, as opposed to joint life annuities

that continue to pay out as long as one of several annuitants remains alive. In addition, we focus on

individuals who purchased annuities at the modal age for men (age 65). We exclude annuitants who died

before their 66th birthday and consider only mortality after age 66, so that we have a uniform entry age.

Our final sample consists of 12,160 annuitants of whom 1,216 are women; this represents about a third of

the single-life sample of all ages analyzed in Finkelstein and Poterba (2004).

We estimate the survival curves for two underlying, unobserved risk types H and L. Our approach, in

the spirit of Heckman and Singer (1984), is to assume a parametric form for the baseline mortality hazard,

and to jointly estimate the parameters of the baseline and the two multiplicative parameters that capture

the unobserved heterogeneity. We follow the actuarial literature on mortality modeling, such as Horiuchi

and Coale (1982), and assume a Gompertz functional form for the baseline hazard. This is particularly

well suited to our context because our data are sparse in the tails of the survival distribution. Formally,

for a given risk typeσ , the mortality hazard at age ix is given by:

(10) ))(exp()( bxx ii −⋅= βασμ σ ,

where b is the base age, 65 in our case. We assume that the growth parameter β is common to both risk

types and to both genders. This means that β determines the shape of the mortality curves for both types,

18

which differ only in the values of σα . Using the notation bxt ii −= , this form of the hazard implies

risk-type-specific survival function of the form:

(11) ⎭⎬⎫

⎩⎨⎧

⋅−= ))exp(1(exp)( ii ttS ββα

σ σ .

When the two underlying risk types are the same for males and females, so that only the mix of these two

risk types is allowed to differ across genders, our stochastic model depends on a parameter vector Θ =

{ Lα , Hα , β , fλ , mλ }. The likelihood function in this case will be:

(12)

( ) ( )

},{)),,|()1()(,|(

)1(1)1(1)(

LHtddtSlwhere

llllL

iiii

i

Lif

Hifm

Lim

Himm

=−+=

−+⋅+−+⋅≡Θ ∑

σβαμβα

λλλλ

σσσσ

In (12), the variable id is an indicator for whether the individual observation is censored and 1m and

1f are indicator variables for whether an individual is male or female respectively. An individual’s

contribution to the likelihood function is a weighted average of the likelihood function of a high risk and

low risk type, with the weights equal to the gender-specific fraction of high and low risk individuals.

Eighty-one percent of the observations in our sample are censored because the annuitant is still alive at

the end of the sample period, December 31, 1998.

Table 1 presents our estimates of the mortality model in (11) and (12). Our estimates yield aggregate

mortality statistics that are similar to those published by the Institute of Actuaries (1999) for all 65 year-

old U.K. pensioners in 1998. For example, the life expectancies implied by our model differ from those

in the aggregate tables by only 0.26 years for women and 0.45 years for men. The estimates of the

mortality rates for the high risk and the low risk types are quite far apart, implying large differences in life

expectancies. For example, the estimates in Table 1 imply that life expectancy at 65 is only 8.8 years for

low risk types, compared to 23.2 for high risk types. Column 5 indicates that over 80 percent of women

are classified in the high risk (long-lived) group, compared to only about 60 percent of men (column 4).

As a result, the estimates imply a 3-year difference in life expectancy at 65 for men compared to women.

19

Survival differences this substantial imply the potential for unisex pricing restrictions to accomplish

considerable redistribution toward the longer-lived women.

We investigated whether the five-parameter model that we estimate is unnecessarily restrictive by

estimating a more flexible eight parameter model that allows for the types to differ across gender. Here,

in addition to having a gender specific fraction of high risk types, λ , the parameters Lα , Hα , and β are

also permitted to be gender specific. Table 2 shows the results. For men, the estimates of the mortality

parameters look qualitatively quite similar to the estimates in Table 1. This is not surprising, since most

of the sample is male. The estimates for women indicate a single underling type for women is the best fit

for the data. In this case, however, the likelihood function for women varies very little as the model

parameters are changed. This explains why we cannot reject the validity of the implicit parameter

restrictions involved in using the 5-parameter instead of the 8-parameter model, as indicated by the a

likelihood ratio test shown in Table 1, column 7 (p=.59). In light of these results, we use the parameter

estimates from our more parsimonious model.

4. Measuring the Efficiency and Distributional Effects of Banning Gender-Based Pricing

This section briefly describes the measures that we use to quantify the efficiency and distributional

effects of a ban on gender-based pricing in the model described above. Standard measures of the

distributional effects of and the efficiency costs of regulatory policies, such as compensating variation,

equivalent variation, and their corresponding measures of deadweight burden, do not naturally extend to

settings with asymmetric information. It is not clear what it means to estimate the transfer that a

consumer of a given type requires to be as well off after a policy intervention as beforehand when it is not

possible for the government to identify this consumer and carry out the transfer. With this consideration

in mind, we develop a measure of inefficiency that is in the spirit of Debreu (1951, 1954). It is also the

natural quantification of the efficiency notion used by Crocker and Snow (1986) when they demonstrate

that restrictions on categorical pricing in insurance markets are efficiency reducing.

20

To construct our efficiency and distribution measures, we use the “actuarial cost function” )(ACσ

from (5), which gives the expected cost to an insurance company of honoring contract A when it is owned

by an individual of risk type σ . The actuarial cost of honoring a vector σ,iA of contracts for each type

},{ YXi∈ and category },{ LH∈σ is given by the total actuarial cost function:

(13)

( ) ( )( )( ) ( )( ),)(1)()1()(1)(

)()1()()(,,,,

,,,

LXLX

HXHX

LYLY

HYHY

XXYYi

ACACACACATCATCATC

λλθλλθθθ σσσ

−+−+−+≡−+≡

where the total cost functions for each category, XTC and YTC , are defined implicitly, and σ,YA and

σ,XA denote category-specific vectors of contracts. The minimum expenditure function is defined by:

(14) ( )⎪⎩

⎪⎨

⎧

∈∀∈∀≥∈∀∈∀≥≡

},{},{),(),~(:)(},{',},{),~(),~(:)(

)~(

',,

',,

,

}~,~,~,~{,

,,,,

LHandYXiSAVSAVMULHandYXiSAVSAVIC

andtoSubject

ATCMinAE

iii

ii

i

AAAAi

HYHXLYLX

σσσ

σσσσσ

σσσσσσ

σ

σ

The minimum expenditure function maps a proposed allocation σ,iA of contracts to each type within

each category into the minimum total actuarial cost of ensuring that each type within each category is at

least as well off as with σ,iA , while respecting the inherent informational constraints in the economy.

These inherent constraints are captured by (IC) in (14), which requires that within each category,

individuals need to be willing to choose the contract A~ designed for them. Because category is

observable, however, incentive compatibility does not have to be satisfied across categories.

An efficient allocation σ,iA solves (14). Any other informationally feasible contract set σ,~ iA that

makes each individual as well off as σ,iA has at least as high a total actuarial cost. Other allocations are

inefficient, and a measure of the inefficiency is )()( ,, σσ ii AEATC − . If σ,1iA and σ,

2iA denote any two

vectors of contracts, the efficiency cost of moving from former to the latter, ),( ,2

,1

σσ ii AAEC is given by

(15) ),( ,2

,1

σσ ii AAEC ) ( ) ( ))()()()( ,1

,1

,2

,2

σσσσ iiii AEATCAEATC −−−≡

21

For our analysis of the policy of banning the use of categorical pricing, this expression simplifies because,

by assumption, the market outcome prior to the ban is efficient. Hence, the efficiency cost of a ban is

exactly the inefficiency of the equilibrium contract set that obtains after the ban.

Both )(⋅TC and )(⋅E decompose by category, so the efficiency cost of a ban on characteristic-based

pricing can be decomposed into category-specific efficiency costs. That is, we can write

)()()( ,,, σσσ iiiiii AcyInefficienAEATC += . This decomposes the actuarial cost, or the resource use, of

a given category into two components: the minimum resources needed to make the types that well off,

and the resources that are wasted because of an inefficient allocation. We interpret the former as a

money-metric measure of the well being of the category, since the wasted resources do not contribute to

well being. We can therefore quantify redistribution at the category level from a policy that changes the

contract set from σ,1iA to σ,

2iA as the increase in this money metric measure. Redistribution towards

category Y is therefore given by ( ) ( ))()(, ,1

,2

,2

,1

iYYiYYiiY AEAEAAR −≡σσ . There is a similar expression

for the redistribution towards category X.

When a policy change has efficiency consequences, the weighted sum across categories of the

redistributions will not be zero, even when the policy change leaves the total actuarial cost unchanged.

This is because some of the redistribution away from category X can be dissipated via an increase in the

inefficiency of the allocations and might never reach category Y. It is perhaps more appealing to have a

redistribution measure in which the entire amount redistributed away from one group is, in fact,

redistributed to the other group. We therefore focus on the re-centered measure:

(16) ( ) ( ) ( ) ( )( )σσσσσσσσ θθ ,2

,1

,2

,1

,2

,1

,2

,1 ,)1(,,,~ iiXiiYiiYiiY AARAARAARAAR −+−≡ .

This measure expresses the re-centered redistribution per member of category Y.

Figure 2 can be used to qualitatively illustrate the efficiency and distributional measures when

category is perfectly predictive of type (i.e., YX λλ −== 10 ). In this setting, the efficiency metric boils

down to summing the certainty equivalent consumptions across types. Prior to the ban, the competitive

22

market gives actuarially fair full insurance contracts *LA and *HA to the two types; this allocation, which

entails state-independent consumption, is efficient. When categorical pricing is banned, the market

implements a pair of contracts labeled LA and HA which is as efficient as it can be, given the

government imposed pricing constraints. This set of allocations is nevertheless inefficient because LA

could, in principle, be replaced by the state independent (full insurance) consumption contract LA′ which

makes L types equally well off, while saving resources. The efficiency cost of the ban is precisely the

difference in the actuarial costs of LA and LA′ , scaled by the number of L types in the market.

The policy also re-distributes resources from L types to H types. The amount redistributed to each of

the H types, computed without re-centering, is the actuarial difference between HA and *HA computed

using mortality risks for H types. We measure the amount redistributed away from each of the L types

via the actuarial difference between *LA and LA′ , in this case computed using the mortality rates for

type L. The change in actual resource use or in the actuarial cost of the L types’ contract is measured by

the actuarial difference between *LA and LA , again using L type mortality rates.

When categorization is imperfect, the same sort of analysis applies, but summing certainty

equivalents across individuals is no longer a valid measure of efficiency. Because contract outcomes are

constrained efficient when categorical pricing is allowed (by assumption), we need only consider the

inefficiency of the post-ban equilibrium. Figure 5 illustrates this. The post-ban allocation is given by the

contract pair HHYHX AAA ≡= ,, and LLYLX AAA ≡= ,, . This allocation is inefficient because of the

inefficient allocation within the X category. Having fewer H types within that category means that

additional (break even) cross subsidies from L types to H types within that category can make both X

category types better off. Hence, both X category types could be made at least as well off with fewer

resources, for example via the pair of contracts indicated in Figure 5. On the other hand, because the Y

category has a greater fraction of H types, additional cross subsidies within that category do not yield

Pareto improvements – the original contracts are, in fact, the efficient way for Y category types to achieve

23

their original level of well being. The efficiency cost of the ban is measured by the difference in the

actuarial costs of the market allocations and the associated efficient allocations.

Because we consider the set of constrained Pareto efficient market outcomes, there is a range of

possible market allocations both prior to and subsequent to a ban in gender-based pricing. As a result,

there is a range of possible estimates of the consequences of a ban. The efficiency and distributional

measures developed above have the nice property that we can summarize all possible efficiency and

distributional effects of a ban via a single-parameter family of consequences. This family ranges from a

“high efficiency cost, low redistribution” end-member to a “low efficiency cost, high redistribution” end-

member. To see this, note that prior to a ban in gender based pricing, the market is, by assumption,

efficient. The efficiency cost of a ban is therefore equal to the inefficiency of the post-ban allocation.

Moreover, because the market does not implement across gender cross-subsidies in the absence of a ban,

the total “welfare” (viz (14)) of each gender prior to the ban is equal to W. The distributional

consequences can be measured via the “welfare” of each gender in the allocation which obtains when a

ban is implemented, regardless of the specifics of the market allocation in the absence of a ban.

The range of possible efficiency and distributional consequences of a ban in gender-based can

therefore be computed from the range of possible market outcomes when a ban is in place – i.e., by the

solutions to (6) as HV varies from the utility )(WV H they get from their full insurance actuarially fair

contract to the utility ( )λAV H they get from a pooled (across gender and type) fair full-insurance

contract. Furthermore, one can show that the redistribution towards women is monotone increasing in

HV and that the efficiency cost is strictly decreasing in HV until the efficiency cost reaches zero and

remains there. Hence, bounding the possible efficiency and distributional consequences of a ban amounts

to computing the solution to (6) at the two endpoints, where the lower end of this range corresponds

precisely with the MWS equilibrium, and the upper end corresponds with the pooled-fair full-insurance

outcome. While this leaves a potentially large range of consequences, it has the advantage of

characterizing the full set of feasible constrained-efficient outcomes. Readers who are willing to choose a

24

particular equilibrium concept – such as the MWS equilibrium – can narrow the range of possible

consequences to a single point.

5. Estimates of the Efficiency and Distributional Consequences of Banning Gender-Based Pricing

We begin by reporting findings for our baseline model, in which firms have full flexibility in

designing the payment profile of the annuities they offer, individuals can save out of their annuity income,

and insurance companies cannot observe saving. After presenting these baseline results, we consider

results in several restricted models and then evaluate the sensitivity of our findings to changing several

key parameters in our analysis.

5.1 Baseline Model Results

To characterize the entire range of possible consequences of a ban in gender based pricing, we need

only to compute two possible post-ban allocations: the MWS equilibrium and the pooled-fair full

insurance outcome. Without loss of generality, we normalize retirement wealth to W = 1 for these

calculations.

Table 3 summarizes the results associated with both the MWS and the pooled-fair outcome, with the

latter labeled SS. The first six columns of Table 3 present the minimum expenditure functions for

women, men, and the total population at each of the two extreme contracts which may obtain when

categorization is banned. These are FE , ME , and E , in the notation used above (see (14)). They denote

the minimum per person resources needed to ensure that each type is at least as well off as in the

equilibrium while respecting the inherent informational constraints of the model. Since each person in

endowed with one unit of resources, the difference between the fifth and sixth columns and 1.0 gives the

efficiency cost of the ban when the post-ban contracts are given by the MWS and are given by the pooled

fair outcomes, respectively. This difference is reported, in percentage terms, in the seventh and eight

columns. For a risk aversion coefficient of 1, the high-end (MWS-end) efficiency cost is 0.04 percent of

retirement wealth W. For risk aversion coefficients of 3 and 5, the comparable costs are about 0.02

25

percent. If, subsequent to a ban, the market implements the pooled fair endpoint outcome, then there are

no associated efficiency costs. It is important to recognize that the small upper bound on the efficiency

costs is largely due to our focus on a compulsory annuity market, and that the efficiency costs of

eliminating characteristic-based pricing in voluntary insurance markets could be many times greater than

our estimates suggest.

The eleventh and twelfth columns of Table 3 report our summary statistics for redistribution from

men to women. This is the re-centered redistribution per woman defined in (16). For a risk aversion of 1,

we estimate that 2.1 percent of the endowment is redistributed when the market implements the MWS

endpoint outcome subsequent to a ban in gender-based pricing. For risk aversion coefficients of 3 and 5,

the comparable numbers are 3.4 percent and 4.1 percent, respectively. The last column of Table 3 reports

the efficiency costs as a percentage of the amount of redistribution for the high-end MWS case. This ratio

varies from 3.6 percent for a risk aversion of 1 to under 1 percent for a risk aversion of 5.

When the market implements the pooled-fair outcome instead, it redistributes a total of 7.14% of

resources towards women. This is between 1.8 and 3.4 times more redistribution than the low-end

redistribution estimates of Table 3. In addition to providing an endpoint for the possible consequences of

a ban in gender-based pricing in our setting, the 7.14 percent redistribution and zero-efficiency cost

endpoint is also interpretable as the effect of banning gender-based pricing in a compulsory full-insurance

setting such as the U.S. Social Security system. In such a setting individuals are, in effect, required to

purchase level inflation-protected annuities with their retirement accumulations W. If categorization by

gender is allowed and pricing is actuarially fair, men get larger per-period annuity payouts than women

for a given initial premium. If categorization is not allowed, all buyers receive the same full insurance

annuity with an intermediate payout level. Because there is no scope for insurers to adjust the menu of

policies that they offer in response to the ban, such a ban would not have any efficiency costs. The

consequences in such a setting are thus identical to the high-distribution endpoint calculations in Table 3.

The smaller redistributive effect of eliminating gender-based pricing in the MWS-endpoints in Table

3, relative to the “Social Security” setting, is a result of the endogenous adjustment of optimal annuity

26

profiles, not of reduced demand for annuities by men, since annuitization is mandatory even in our

benchmark setting. The reduction in redistribution results from the fact that firms can sell annuity

contracts that vary in the time profile of their payout stream and that, by using these profiles for screening

purposes, they can partially undo the transfers that take place as a result of the ban on gender-based

pricing. This highlights how recognition of the endogenous structure of insurance contracts to

government regulation can have important effects on analyses of the regulatory policy.

5.2 Results in Restricted Models

We compare the results from our baseline model with those from two alternative models. The first

restricts the behavior of annuity buyers by disallowing saving, and the second restricts the behavior of

annuity providers by limiting the space of contracts they can offer. These exercises serve two related

purposes. First, they help to expand our understanding of how various provisions in our model affect our

results. Second, they illustrate the importance of extending the basic model to account for such real-

world features as access to savings or limits on the set of contracts insurers can offer. In both cases, we

focus exclusively on the high-efficiency cost low-redistribution endpoint, since the other endpoint is

unaffected by these changes.

Table 4 summarizes the results of with each of these generalizations. We explained earlier that if

annuitants cannot save, or if their saving can be observed and contracted upon by insurance companies,

then the MWS equilibrium annuities of short-lived types are characterized by contracts that are level until

very old ages, at which point payments fall off quite rapidly. Because long-lived types have a substantial

chance of being alive at those old ages, relative to the short-lived types, this shape enforces self-selection

at very little cost to the short-lived types. In practice, this means that the MWS equilibrium contracts

offered to each sub-population, whether males alone, females alone, or the pooled population, involve

zero cross-subsidies from the short-lived to the long-lived types, and the MWS equilibrium coincides with

the RSR equilibrium. Bans in categorization have neither efficiency nor distributional consequences in

this setting.

27

In contrast, restricting the set of contracts that insurers can offer can increase the efficiency costs of a

ban on gender-based pricing while reducing the amount of redistribution. This restriction is imposed to

more closely accord with the payment profiles of policies actually observed in the U.K. annuity market.

While annuity companies appear to use the time-profile of annuity payments to screen individuals

according to their risk type in the United Kingdom, Finkelstein and Poterba (2002, 2004) report that

insurers offer only a limited number of simple alternative payment profiles. Most policies involve level

nominal payments; the majority of the remainder involve nominal payments that escalate at a constant

rate over time. The declining annuities generated by our baseline model do not have this feature. It is

possible that a richer and more realistic model might yield annuities with a structure that more closely

accords with observed policies. Another possibility is that there are some implicit restrictions on the

form of annuities that can be offered by insurance firms. Such limitations might arise, for example, if

there are fixed costs of offering different insurance products, explicit or implicit regulations on legal

pension payment profiles, or costs to either the consumer or producer from product complexity.

The particular restriction we consider limits firms to offering only policies which provide benefits

that rise or fall at a constant real rate: tt aa η=+1 for some constant η and for all t. Subject to this

additional requirement, market outcomes are still characterized by (6). As in the unrestricted program,

the long-lived types purchase a full-insurance annuity, and short lived types purchase a declining annuity.

For the baseline parameters and a risk aversion of 3, the MWS equilibrium rate of decline is 12.1 percent

per annum when gender-based pricing is banned, and is 9.5 percent and 13.3 percent for short-lived males

and females, respectively, when gender-based pricing is allowed. Table 4 indicates that for a risk

aversion of 3, a ban in gender-based pricing in this redistricted contract model redistributes approximately

2.25 percent of retirement wealth towards women, at an efficiency cost of 0.136 percent of retirement

wealth. Compared with the results in the baseline model without contract restrictions the maximum

amount of redistribution achievable by a ban on gender-based pricing falls by about one-third in a model

with contract restrictions; the efficiency costs, while still modest on an absolute scale, rise by an order of

28

magnitude. These findings highlight how the nature of the contracting environment and the potential

endogenous response to regulation can have substantial effects on the consequences of regulation.

These results also provide insight into why the efficiency costs are so small in the baseline model.

There are two mechanisms for satisfying self-selection constraints in an MWS equilibrium. First, the

short-lived (L) types can be offered a highly distorted contract, such as a contract with front loading. This

distortion makes the L type contract less attractive to both types, but it is a distortion which is

differentially more unattractive for the H types. Second, there can be cross-subsidies from the L types’

contracts to the H types’ contracts. These help satisfy self-selection by making the H type annuity

contracts more desirable and the L type annuity contracts less desirable. The efficiency costs will tend to

be large when a change in the mix λ of H and L types has substantial effects on the optimal amount of

distortion in the contract space.

When it is not possible to save, there is essentially no tradeoff between efficiency and redistribution.

Distortions can be used to enforce self selection at virtually no costs, so the equilibrium never relies on

cross-subsidies. This in turn means that there is no change in the distortion when a ban is put in place,

and therefore no efficiency cost. More generally, whenever the marginal costs of distortion are very small

for low distortions, and very high at high distortions – with a sharp transition between these two regions –

the efficiency costs of a ban will tend to be low, as the optimal mix of distortion and cross-subsidization

will take place near the transition, irrespective of the relative fraction of low and high-risk types.

Restricting the contract space raises the efficiency cost of a ban on gender-based pricing because the

transition is not as sharp in the restricted contracts case. With an unrestricted contract space, it is possible

to target an optimal distortion, for example, by making the L type annuity more downward sloping at old

ages than at young ages. This flexibility means that the first bit of distortion is the most useful, and

additional distortions quickly become less and less useful. In contrast, with the restricted contract spaces

we consider, the distortion cannot be targeted: the size of the distortion is fully captured by the downward

tilt of the L type annuity. Relative to the unrestricted space, the tradeoff between distortion and cross-

subsidy is therefore flatter, making the efficiency cost of banning category-based pricing higher.

29

5.3 Comparative Statics

To provide some insight into the sensitivity of our results to various parameters, we computed the

amount of redistribution and the efficiency cost of banning categorization under three alternative sets of

parameter vectors. Table 5 reports the results, again for the high-efficiency low-redistribution endpoint.

First, we vary the fraction θ of women in the population. Our base case in Table 3 assumed a 50-50

gender split. Decreasing θ, to reflect the fact that most participants in the compulsory U.K. annuity

market are male, increases the per-woman distributional effects of banning categorization. When there

are relatively more men, women gain more by being pooled with the men.

The efficiency cost of a ban, however, is non-monotonic in θ . A change in θ has two offsetting

effects on efficiency. First, the efficiency costs mechanically fall as the relative size of the male

population decreases, since the efficiency costs of a ban in categorization in the MWS framework are

entirely due to the inefficiency of the post-ban allocation amongst the low risk category, which in this

case is men. Second, as the number of women increases, the non-categorizing equilibrium payout moves

away from the men’s categorizing payout and toward the women’s. This raises the efficiency cost per

male, and thus creates an effect that operates against the mechanical first effect. Finkelstein and Poterba

(2004, 2006) suggest that about 70 percent of U.K. annuitants are male. The results in Table 5 suggest

that this raises the amount of redistribution to women and decreases the efficiency cost per dollar of

redistribution by about 40 percent compared to our baseline estimates based on the 50-50 gender split.

The second comparative static we consider involves varying the pair Hα and Lα , the mortality

hazard at retirement for the two different risk types. We vary these two in a way that keeps the

population average mortality hazard approximately constant at retirement age. The gap between the two

risks types in our baseline parameterization may be too large, since, at best, our estimates describe the

differences in actual risks across types, as opposed to the private information individuals have when they

make annuity purchases. As the hazard rates move closer together, the amount of redistribution that takes

place as a result of the ban decreases. The total efficiency cost, however, appears to be robust to the gap

30

in the mortality rates. As a result, the efficiency cost per dollar of redistribution rises as the relative hazard

declines.

The final variation we consider is jointly varying Hα and Lα – the age 65 mortality hazards for the

two types – and the gender-specific fractions of each risk type, Mλ and Fλ , in such a way that life

expectancies of the two genders remains constant and the aggregate fraction of high risk and low risk

types remains unchanged. This is accomplished by first varying Hα and Lα so as to keep aggregate life

expectancy constant, and then by adjusting the gender-specific type fractions to keep the life expectancy

of each gender unchanged. Thus, like the previous variation, the thought experiment implicit in this

variation is to change the mortality gap; this way of doing so may be more reasonable than the one above.

Like the previous variation, this has small but non-zero effects on our estimates of the distributional

consequences. With a smaller gap, the distributional consequences are smaller. In contrast with the

previous type of mortality gap variation, however, we see that the efficiency consequences can be

substantially increased by a lowering of the mortality gap. Indeed, for the smallest gap considered, the

efficiency consequences are approximately six times larger than in the baseline case.

6. Conclusion

This paper investigates the economic effects of restricting the set of individual characteristics that can

be used in pricing insurance contracts. It moves beyond the qualitative observation that such regulations

may entail efficiency costs to explore quantitatively both the distributional and efficiency effects of such a

policy. To do so, we develop, calibrate, and solve an equilibrium contracting model for the compulsory

retirement annuity market in the United Kingdom.

Our findings underscore the importance of considering the endogenous response of insurance

contracts to regulatory restrictions when assessing the impact of regulation. Our central estimate suggests

that allowing for such endogenous response may reduce estimates of the amount of redistribution from

men to women under a ban on gender-based pricing by as much as fifty percent. This estimate contrasts

31

the endogenous response case with an alternative in which the menu of policies is fixed, as it is when

governments provide compulsory annuities with fixed payout structures in Social Security programs.

The redistribution associated with a unisex pricing requirement, even accounting for the endogenous

contract response, remains substantial. Our baseline estimates suggest that at least 3.4 percent of

retirement wealth is redistributed from men to women. We also estimate that in the compulsory annuity

setting, unisex pricing rules would impose only a modest efficiency cost, approximately 0.02 percent of

retirement wealth. Recall, however, that our analysis focuses only on the set of individuals who are

already covered by retirement plans that require annuitization of account balances at some point, so non-

participation in the annuity market is not an option for them. Our efficiency estimates almost certainly

understate the efficiency costs of unisex pricing in voluntary annuity markets, since they do not consider

consumer decisions about whether or not to participate in the market.

Our estimates also fail to capture the potential long-run behavioral responses to unisex pricing

regulations. For example, a change in annuity pricing could affect the savings and labor supply decisions

of those who will subsequently face compulsory annuitization requirements. Annuity companies might

also respond to unisex pricing requirements by conditioning annuity prices on other observables that are

not currently used in pricing policies, such as occupation or location of residence. Discussions of gender-

neutral pricing in insurance markets also raise interesting questions that range far beyond our study, such

as why a society might wish to carry out transfers between men and women, the extent to which gender-

based transfers in the marketplace are simply undone within the household, and why insurance markets

rather than, say, the tax system, are a natural locus for such transfers. These are all interesting avenues to

explore in future work.

Restrictions on the use of gender in pricing retirement annuities are just one of many examples of

regulatory constraints on characteristic-based pricing in private insurance markets. Many states in the

United States, for example, restrict insurers’ use of information on the individual’s gender, race,

residential location, or past driving history, in setting automobile insurance rates. Similar restrictions

apply in homeowner’s insurance markets and in many small-group and non-group health insurance

32

markets. Moreover, the growing field of medical and genetic testing promises to create new tensions

between insurers and regulators, as medical science provides new information that insurers could

potentially use to predict the future morbidity and mortality of potential clients for life and health

insurance policies.

The framework we have developed provides a natural starting point for evaluating the efficiency and

distributional consequences of current or potential future restrictions on characteristic based pricing in

these other markets. Such evaluations also raise several new issues which we did not have to confront in

the case of unisex pricing requirements for annuities. In the setting we analyze there is scope for choice

and self-selection on some of the dimensions of the annuity contract but not on the extensive margin of

whether or not to annuitize at all. In addition, while moral hazard is likely to be relatively unimportant in

the annuity market, the moral hazard effects of automobile or health insurance may be more pronounced,

and will need to be considered in analyzing the efficiency consequences of regulatory restrictions. Finally,

gender is an immutable characteristic, unlike geographic location or past driving records, and will

therefore not change endogenously in response to the pricing regime. The endogenous adjustment of

characteristics to the pricing regime is another interesting issue that future work should consider.

33

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Annuities: New Evidence from the United Kingdom,” Economic Journal 112, 28-50. Finkelstein, Amy and James Poterba (2004). “Adverse Selection in Insurance Markets: Policyholder

Evidence from the U.K. Annuity Market,” Journal of Political Economy 112, 183-208. Finkelstein, Amy and James Poterba. (2006). “Testing for Adverse Selection with Unused

Observables” Unpublished mimeo. Heckman, James and Burton Singer (1984). “Econometric Duration Analysis,” Journal of

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77, 298-312. Hurd, Michael (1989). “Mortality Risk and Bequests.” Econometrica 57, 779-814. Institute of Actuaries. 1999. Continuous Mortality Investigation Reports Numbers 16 and 17. Kotlikoff, Laurence and Lawrence Summers (1981). “The Role of Intergenerational Transfers in

Aggregate Capital Formation” The Journal of Political Economy 89, 706-732. Miyazaki, Hajime (1977), “The Rate Race and Internal Labor Markets,” Bell Journal of Economics

8, 394-418.

34

Polborn, Mattias, Michael Hoy, and Asha Sadanand (2006). “Advantageous Effects of Regulatory Adverse Selection in the Life Insurance Market.” Economic Journal 116, 327-354.

Posner, Richard (1971), “Taxation by Regulation,” Bell Journal of Economics 2 (1), 22-50. Rothschild, Michael and Joseph E. Stiglitz (1976), “Equilibrium in Competitive Insurance Markets:

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Simon, Kosali. (Forthcoming). “Adverse Selection in Health Insurance Markets: Evidence from State Small-group health insurance reforms.” Journal of Public Economics

Spence, Michael (1978), “Product Differentiation and Performance in Insurance Markets,” Journal of Public Economics 10 (3), 427-447

Wilson, Charles (1977). “A Model of Insurance Markets with Incomplete Information,” Journal of Economic Theory 16, 167-207.

35

Figure 1: Stylized constrained Pareto frontier. The dark curve connecting M to F depicts a portion of the locus of the L-type consumption points at constrained Pareto optimal outcomes. Specifically, it depicts the L-type consumptions consistent with:

(i) H-types receiving full insurance consumption; (ii) The H-types incentive compatibility constraint binding; (iii) Firms breaking even on aggregate, and (iv) H-types being no better off than at F.

A symmetric portion of this frontier (corresponding with even larger cross-subsidies from L- to H-types) lies on the other side of the full insurance locus.

Consumption if “no accident”

F

M

)(WVV HH =

( ))(),( λAVWVV HHH ∈

)( λAVV HH =

Full Insurance Locus

Con

sum

ptio

n if

“acc

iden

t”

Pooled Price Actuarially Fair Locus

Endowment Point

H-Risk Iso-Cost Lines

L-Risk Iso-Cost Lines

L-Risk Indifference Curve

H-Risk Indifference Curves

45o

36

LA

State 0 Consumption

Sta

te 1

Con

sum

ptio

n

W

*LA

*HA

( )*LLL AVV = ( )LLL AVV =

( )HHH AVV =

L type Isocost Lines

H type Isocost Lines

HA

( )*HHH AVV =

Figure 2: A Constrained Efficient Annuity Pair

Full Insurance Locus

LA′

LA

37

Figure 3: MWS Equilibrium Annuities if Savings is Impossible (Male population, 3=γ )

38

Figure 4: MWS Equilibrium Annuities if Unobservable Savings is Possible (Male population, 3=γ )

39

State 0 consumption

Sta

te 1

con

sum

ptio

n

W

Full Insurance Locus

Equilibrium Contracts when Categorization is Banned

Efficient Contracts when Type is Observable

Efficient X Category Contracts when Only Category is Observable

HA LA

Figure 5: The inefficiency of bans in categorical pricing

40

Table 1: Estimates of Two-Type Gompertz Mortality Hazard Model, Same Types for Both Genders Sample Multi-

plicative factor on hazard for high risk (αH)

Multi-plicative factor on hazard for low risk (αL)

Common growth factor in hazard model (β)

Fraction of men who are high risk (λM)

Fraction of women who are high risk (λF)

log(L)

χ2(3) (P-value)

65 Year Olds (N=12160)

0.0031 (0.0003)

0.0405 (0.0013)

0.1485 (0.0056)

0.6051 (0.0096)

0.8192 (0.0231)

-10347.45 1.94 (0.59)

Notes: Results are based on estimating equation (12) using micro-data on annuitant mortality patterns. Standard errors are in parentheses. Column 6 contains the total log likelihood. Column 7 reports the χ2

(3) statistic (P-value) for the Likelihood Ratio test of this restriction relative to the more flexible specification in Table 2. Table 2: Estimates of Two-Type Gender-Specific Gompertz Mortality Model Sample Multiplicative

factor on hazard for high risk (αH,m / αH,f)

Multiplicative factor on hazard for low risk (αL,m / αL,f)

Common growth factor in hazard model (βm / βf)

Fraction who are high risk (λm /λf)

log(L), by gender

log(L)

65 Year Old Males (m) (N=10944)

0.0030 (0.0003)

0.0423 (0.0014)

0.1566 (0.0058)

0.6305 (0.0091)

-9568.59

65 Year Old Females (f) (N=1216)

0.0111 (0.0009)

NA

0.0882 (0.0228)

NA -777.89

-1036.4

Notes: Results are based on estimating equation (12) separately for each gender using the same data as in Table 1 Standard errors are in parentheses. The estimation for females led to a single type model. The final column reports the total log likelihood. Table 3: Range of Efficiency and Distributional Consequences of Unisex Pricing

Required Per-Person Endowment Needed to Achieve Utility Level from Non-Categorizing Equilibrium When Categorization is Allowed

Women ( WE )

Men ( ME ) Total Pop’n ( E )

Eff’cy Cost as % of Total Endowment

Redistribution to Women ( WR~ ), Per Woman, as a % of Endowment

Eff’cy Cost Per Dollar Redistn

Coef’ of Rel’ve Risk Aver-sion

MWS SS MWS SS MWS SS MWS SS MWS SS MWS

γ=1 1.0205 1.0714 0.9788 0.9286 0.9996 1 0.0381% 0% 2.084% 7.14% 3.66% γ=3 1.0336 1.0714 0.9659 0.9286 0.9998 1 0.0246 0 3.387 7.14 1.45 γ=5 1.0404 1.0714 0.9593 0.9286 0.9998 1 0.0180 0 4.055 7.14 0.89

Notes: Estimates are based on the model and algorithm described in the text. Columns labeled MWS refer to the high efficiency cost/low redistribution end of the range of possible consequences which obtains when the market implements the Miyazaki-Wilson-Spence equilibrium when gender-based pricing is banned. Columns labeled SS refer to the zero efficiency cost/high redistribution end of the range which obtains when the market implements a pooled-fair full insurance “Social Security-like” outcome when gender based pricing is banned. The MWS contracts are computed using Equation (6) and the risk type-distributions estimated in Table 1, pooled across genders. Columns (1)-(6) are computed using Equation (15) and columns (9)-(10) are computed using Equation (16).

41

Table 4: Efficiency and Distributional Effects of Ban on Gender Based Pricing in Restricted Models Redistribution to Women ( WR~ ), Per

Woman (as % of Endowment) Efficiency Cost as % of Endowment

MWS SS MWS SS γ=1

Unrestricted (Baseline) Model 2.0838 7.14 0.0381 0 No Savings Model 0 7.14 0 0 Restricted Contracts Model 1.3326 7.14 0.1000 0

γ=3 Unrestricted (Baseline) Model 3.3874 7.14 0.0246 0 No Savings Model 0 7.14 0 0 Restricted Contracts Model 2.2504 7.14 0.1358 0

γ=5 Unrestricted (Baseline) Model 4.0549 7.14 0.0180 0 No Savings Model 0 7.14 0 0 Restricted Contracts Model 2.8690 7.14 0.1352 0 Notes: Unrestricted (Baseline) Model calculations are as in Table 3. The Restricted Contracts Model calculations are described in Section 5.2: in this model, firms can only offer contracts with constant escalation or declination rates. In the No Savings Model, individuals are assumed to have no access to savings technology, as described in Section 5.2.

42

Table 5: Sensitivity Analysis for Redistribution and Efficiency Cost Calculations, (γ = 3) Parameter Being Varied and New Value

Redistribution to Women ( WR~ ), Per Woman (as % of Endowment)

Efficiency Cost as % of Endowment

Efficiency Cost Per Dollar of Distribution

MWS SS MWS SS MWS SS

θ (fraction women) 0.1 6.37% 13.63% 0.00% 0% 0.32% 0%0.3 4.84 10.30 0.01 0 0.89 00.5 3.39 7.14 0.02 0 1.45 00.7 2.00 4.17 0.03 0 1.97 00.9 0.66 1.35 0.01 0 2.40 0

Hα , Lα = Mortality hazard at age 65 for low-risk and high-risk type .001, .046 4.72% 8.63% 0.02% 0% 0.91% 0%.002, .043 3.98 7.85 0.02 0 1.18 0

.0031, .041 3.39 7.14 0.02 0 1.45 0.005, .036 2.62 6.01 0.03 0 1.97 0.008, .028 1.65 4.16 0.03 0 3.27 0

( Hα , Lα ),( λm, λf ): Age 65 mortality hazards and type fractions (0.0021,0.2492),(0.6445,0.7798) 4.69% 7.89% 0.01% 0% 0.38% 0%(0.0026,0.0793),(0.6275,0.7968) 3.82 7.45 0.02 0 0.82 0(0.0031,0.0405),(0.6051,0.8192) 3.39 7.14 0.02 0 1.45 0(0.0036,0.0248),(0.5738,0.8505) 3.09 6.92 0.04 0 2.58 0(0.0041,0.0169),(0.5268,0.8976) 2.86 6.78 0.07 0 4.90 0(0.0046,0.0122),(0.4477,0.9770) 2.64 6.62 0.14 0 10.72 0Note: Same calculations as in Table 3 with varying parameters. Results for baseline parameters from Table 3 appear in bold. The mortality hazards for high and low risk types at age 65 are varied while keeping the aggregate mortality rate at age 65 constant. The mortality hazards and type fractions in the bottom panel are varied to keep aggregate type fractions and gender-specific life expectancies constant.

Appendix: Solution Algorithm for Program (7)

This appendix describes and proves the validity of our procedure for solv-ing Program (7) in the text. The difficult part of solving (7) stems from theneed to compute V H(AL), the utility H types achieve when they purchasethe annuity contract designed for the L types and save optimally. We dealwith this difficulty by identifying the structure of the optimal saving patternof deviating H types at the solution to (7).

There are two key features to this structure. First, deviating H types havean incentive to save only at old ages. There is some period n∗ before whichdeviating H types consume the annuity stream. We can therefore solve forV H(AL) by examining the savings behavior in periods n ≥ n∗ only. Second,deviating H types will optimally carry strictly positive wealth forward atevery date n ≥ n∗. Intuitively, absent savings the (IC ′) constraint in (7)could be satisfied with an annuity stream AL which drops off very steeply atvery old ages. Such an annuity would provide H types with an incentive tosave at old ages, undermining the effectiveness and desirability of the steepdrop off. The ability of H types to save therefore pushes the “drop off” inthe annuity AL to earlier dates than would otherwise be optimal. For thisreason, deviating H types never have incentive to borrow at the optimal AL:if they did, AL could be improved by pushing the “drop off” back towardslater ages.

The first feature is important for us: at the heart of our solution procedureis an algorithm to find the n∗ after which the deviating H type’s begin to dosomething other than just consume the annuity stream. The second featureis important because it makes (7) analytically tractable. To see why, contrastthe indirect utility of deviating H types in two situations. In both, take theirbehavior before n to involve the direct consumption of the annuity streamAL prior to n. The two situations only differ in the potential behavior aftern.

In the first situation, we know nothing about the post-n savings behaviorof H types, so we must solve:

V H(A; n) ≡

maxΓ ≡ {c0, · · · , cN} UH(Γ)

subject to(it) ct = at ∀t < n

(iit)∑t

s=n δs (cs − as) ≤ 0 ∀t ≥ n

(17)

43

to find their utility from a given consumption stream. In the second situation,we know that H types will always choose to carry positive wealth after n.This means that we can instead solve:

V H(A; n) ≡

max(c0, · · · , cN)

UH(c0, · · · , cN)

subject to

(it) ct = at ∀t < n

(ii)∑N

s=n δs (cs − as) ≤ 0

. (18)

Programs (17) and (18) differ in the constraints (iit) and (ii). The formerinvolves one “no borrowing” constraint for each period t ≥ n: the total re-sources consumed through period t cannot exceed the total resources receivedup to that point. In contrast, the latter only has a single “lifetime” resourceconstraint. When we know that H types will always choose to carry positivewealth after n, we know that the no borrowing constraints are slack, and wecan drop all of them except the whole-life no borrowing constraint.

Program (18) is easily solved using first order methods. With constantrelative risk aversion utility, this solution yields a closed-form expression forV H(A; n) and its derivatives. This allows us to solve (7) using first ordermethods once we have identified the cutoff value n∗. We will present ouralgorithm for constructing n∗ below.

Before presenting our algorithm, let us formalize the preceding intuition.Suppose we knew that deviating H types would consume the entire annuitypayment in each period prior to n. Fix a Lagrange multiplier ν on constraint(IC ′) in (7), fix a T for which constraint (MU ′) binds, let V = V H(W +1−λ

λT ), and let W = W − T . Then solving (7) for this fixed ν and T would

be equivalent to solving the program

(Pn)

maxAL

{V L(AL)− ν

(V H(AL; n)− V

)}

subject to(BC ′) CL(AL) ≤ W

.

Solving (7) is always equivalent to solving (P0) for the proper value of ν andT . When we know that deviating H types will consume the entire annuitypayment in each period prior to n, solving (Pn) is equivalent to solving (P0)as well. If we additionally knew that H types would carry strictly positivewealth in every period after n, solving (Pn) would also be equivalent to solving

44

the program:

(Pn)

maxAL

{V L(AL)− ν

(V H(AL; n)− V

)}

subject to(BC ′) CL(AL) ≤ W

.

When we know the two features of deviating H type’s consumption patternsare satisfied and we know the cutoff n∗, solving (Pn∗) will therefore also solve(7). This is important, because the closed, tractable form of V H(A; n) allowsus to solve (Pn) using first order methods.

We will now present Algorithm 1, which we use to construct n∗. Theremainder of the appendix will be devoted to showing that the solutions to(P0) and (Pn∗) coincide for this n∗. This is formally stated in Proposition 1below, but we will need to establish several lemmas before we can prove it.Once we have proved it, we will know that applying Algorithm 1 to find n∗

and then solving (Pn∗) will solve (7) for the given ν, and we will be done.First we define a parameter n∗max which will play an important role in

Algorithm 1. To motivate it, imagine solving (PN) for AL∗ = (aL∗0 , · · · , aL∗

N ).If it happens that

SHn (aL∗

n )−γ ≥ SHn+1(a

L∗n+1)

−γ for n = 0 · · · , N − 1, (19)

then H types will have no incentive to save when given annuity AL∗. Hence,AL∗ will also solve the tighter program (P0). To see when (19) is possible,consider the first order conditions for aL∗

n and aL∗n+1. These imply

(aL∗n )−γ

(1− ν

SHn

SLn

)≥ (aL∗

n+1)−γ

(1− ν

SHn+1

SLn+1

). (20)

Combining (19) and (20) yields

ν ≤( 1

SHn+1

− 1SH

n

1SL

n+1− 1

SLn

). (21)

Therefore, (19) will only be possible—and AL∗ can only solve (P0)—when ν is

sufficiently low. For higher ν, there will be some t for which ν >

(1

SHn+1

− 1

SHn

1

SLn+1

− 1

SLn

),

45

and we will need to solve (P0) using some other method. This motivates thefollowing definition:

n∗max ≡ min

{{N} ∪

{n ∈ {0, · · · , N − 1} : ν ≥

( 1SH

n+1− 1

SHn

1SL

n+1− 1

SLn

)}}, (22)

so that n∗max = N if and only if (19) is possible. If n∗max < N , then we needsome other method for solving (P0). This is is purpose of Algorithm 1.

Algorithm 1

1. Start with n = n∗max.

2. If n = 0 or if SHn−1

(cnn−1

)−γ> SH

n (cnn)−γ, stop, n∗ = n. Otherwise, take

n = n− 1 and repeat step 2.

Algorithm 1 starts with n = n∗max and solves (Pn∗max) for An∗max . It checks

if H types have a (weak) incentive to save at n∗max−1 given their optimal con-sumption pattern when given An∗max—i.e., the consumption vector Γ solving(17) defining V H(An∗max ; n∗max). If not, stop. If so, decrement n and repeatusing n instead of n∗max, continuing to decrement n until either there is noincentive to save at n− 1, or until n = 0.

Our first lemma shows that the date n∗max is the cutoff n between ν >(1

SHn+1

− 1

SHn

1

SLn+1

− 1

SLn

)and ν <

(1

SHn+1

− 1

SHn

1

SLn+1

− 1

SLn

). This plays a key role in assuring that

the algorithm works correctly.

Lemma 1 For the Gompertz mortality curves we consider,

(1

SHn+1

− 1

SHn

1

SLn+1

− 1

SLn

)is

declining in n.

Lemma 1 is easily verified by numerical computations for our particular para-metrization of the Gompertz mortality curves. A formal proof of the lemmafor any pair of Gompertz mortality curves involves tedious algebra and alimiting argument. It is omitted here but is available upon request from theauthors.

Our second lemma characterizes the consumption patterns Γn = (cn0 , · · · , cn

N)which solve (17) for a given solution An = (an

0 , · · · , anN) to (Pn). Note that,

by assumption, any such consumption pattern has cnt = an

t for t ≤ t0.

46

Lemma 2 If An = (an0 , · · · , an

N) solves (Pn), and Γn = (cn0 , · · · , cn

N) solvesthe program defining V H(An; n), then ∃ an integer k ≥ 0 and a set T ={t0, · · · , tk, tk+1} of integers ti, with t0 ≡ n−1, ti < ti+1, and tk+1 = N , suchthat:

• For t0 < t < t′: SHt (cn

t )−γ ≥ SHt′ (cn

t′)−γ , with equality iff ∃i such that

ti < t and t′ ≤ ti+1; and

• For each i ≤ k,t∑

t=ti+1

δn (cnt − an

t ) ≤ 0,

for each ti + 1 ≤ t ≤ ti+1, with equality if t = ti+1.

Lemma 2 states that the dates after n − 1 can be broken up, by some setof cutoff values T, into a series of intervals [ti + 1, · · · , ti+1]. Within eachinterval, H types consume in such a way that they have no incentive to saveor borrow. At the upper end ti of an interval, the H type’s consumption issuch that they have a strict incentive to shift consumption from ti + 1 backto ti; they cannot do so, because they cannot borrow and they do not carrypositive wealth between ti and ti + 1. The “proof” involves simply lookingat Cn and An and defining the appropriate set T.

Lemmas 3, through 6 below characterize the cutoff values T for solutionsto (Pn). Specifically, Lemma 3 presents some first order necessary conditionsfor solving (Pn). Lemma 4 uses these first order conditions to establishsome properties of the annuity and consumption streams associated with thesolution to (Pn), taking the set of cutoffs T as given. Lemma 5 establishesthat when the solution to (Pn) involves the cutoffs T = {n− 1, N}, it is alsoa solution to (Pn). Lemma 6 then uses the properties of Lemmas 3 and 4 toshow that the only set T consistent with solving (Pn) when n∗ ≤ n ≤ n∗max isthe (minimal) set {n− 1, N}. Together, these will tell us that the solutionsto (Pn∗) and (Pn∗) coincide, which enables us to prove Proposition 1.

Lemma 3 Let An ≡ (an0 , · · · , an

N) solve (Pn), let Γn = (cn0 , · · · , cn

N) solve theprogram defining V H(An, n), and let T = {t0, · · · , tk, tk+1} be the associatedset of integers from Lemma 2. Let µ be the Lagrange multiplier associatedwith the constraint (BC ′). Then the following must hold:

µ = (ant )−γ − (cn

t )−γ νSH

t

SLt, ∀t ∈ {0, · · · , N}, (23)

47

ant = cn

t , ∀t < N, (24)

SHt (cn

t )−γ = SHt′ (cn

t′)−γ, ∀t, t′ ∈ {ti + 1, · · · , ti+1} ∀i ∈ {0, · · · , k}, (25)

∑ti+1

t=ti+1 δt(cnt − an

t ) = 0, ∀i ∈ {0, · · · , k}. (26)

Proof. Since ∂V H(An;n)∂an

t= SH

t (cnt )−γ, (23) is the first order necessary con-

dition for ant in (Pn). Conditions (24)-(26) characterize necessary conditions

for Γn to solve the program defining V H(An; n). Condition (24) follows fromthe definition of that program. Both (25) and (26) follow from Lemma 2:(25) states that H type’s consumption is such that they have no incentiveto borrow or save within an interval and (26) states that individuals do notcarry positive wealth from one interval to the next.

By Lemma 3, conditions (23)-(26) are necessary for a solution to (Pn).Lemma 4 shows that for any fixed set of cutoffs T, these four conditions aresatisfied only for a unique annuity and consumption pair. The lemma furtherexamines how this unique pair varies with the Lagrange multiplier µ: sinceµ can be interpreted as a marginal utility of resources and u′(x) = x−γ, the

pair varies with µ as µ−1γ .

Lemma 4 Fix µ > 0 and T. Then there is a unique annuity and con-sumption pair, (a′n0 , · · · , a′nN) ≡ A′n and (c′n0 , · · · , c′nN) = Γ′n, that solves

(23) through (26). Viewed as a function of µ, a′nt (µ) = a′nt (1)µ−1γ and

c′nt (µ) = c′nt (1)µ−1γ .

Proof. Fix a ti. Condition (25) determinesc′ntc′nt′

for any t, t′ in the interval

[ti+1, · · · , ti+1]. (c′nti+1, · · · , c′nti+1) is therefore determined up to a scalar multi-

ple. To pin down this scalar multiple, fix a Wi ∈ R and generate the uniquevector (c′nti+1, · · · , c′nti+1

) consistent with (25) and with Wi =∑ti+1

t=ti+1 δtc′nt .Next, define the function M1 : R→ Rti+1−ti by M1(a

nti+1) ≡ (an

ti+1, · · · , anti+1

),where an

t is defined implicitly via

(ant )−γ − (c′nt )

−γνSH

t

SLt

=(an

ti+1

)−γ − (c′nti+1

)−γνSH

ti+1

SLti+1

,

as required by (23). Similarly, define the function M2 : Rti+1−ti → R viaM2(a

nti+1, · · · , an

ti+1) ≡ ∑ti+1

t=tiδtan

t . Then M2(M1(ati+1)) is strictly increasing

48

in ati+1; hence there is a unique ati+1 such that M2(M1(ati+1)) = Wi. There-

fore, for any Wi, there is a unique pair of vectors(a′nti+1(Wi), · · · , a′nti+1

(Wi))

and(c′nti+1(Wi), · · · , c′nti+1

(Wi))

consistent with

Wi =

ti+1∑t=ti+1

δta′nt (Wi) =

ti+1∑t=ti+1

δtc′nt (Wi)

and with

(a′nt (Wi)

)−γ

−(c′nt (Wi)

)−γ

νSH

t

SLt

=(a′nti+1(Wi)

)−γ

−(c′nti+1(Wi)

)−γ

νSH

ti+1

SLti+1

for all t ∈ {ti + 1, · · · , ti+1}.Clearly, if

{(a′nt (Wi)

)ti+1

t=ti+1,(c′nt (Wi)

)ti+1

t=ti+1

}is the unique pair consis-

tent in this sense with Wi, then

{(βa′nt (Wi)

)ti+1

t=ti+1,(βc′nt (Wi)

)ti+1

t=ti+1

}is

uniquely consistent in this sense with βWi for any β > 0. Via µ, (23) then

pins down a unique Wi and corresponding(a′nti+1(Wi), · · · , a′nti+1

(Wi))

and(c′nti+1(Wi), · · · , c′nti+1

(Wi))

consistent with (23), (25) and (26) for the inter-

val i, and shows that c′nt and a′nt vary with µ as µ−1γ in this interval.

This argument holds for each ti, and hence for each t ≥ n. For t < n, asimilar argument using (24) instead of (25) establishes the same uniquenessresult, completing the proof.

Lemma 4 shows that there is a unique pair An and Γn that satisfies thenecessary conditions for a given fixed T. That is, for any T there is a unique“candidate” for solving (Pn). We will now establish two lemmas about thiscandidate solution. First, Lemma 5 shows that if the candidate associatedwith cutoffs T = {n − 1, N} is indeed a solution to (Pn), then it is also asolution to (Pn). Second, Lemma 6 shows that, when n∗ ≤ n ≤ n∗max, thecandidate for any other T = {n−1, N} cannot solve (Pn) for T = {n−1, N}.Together, they imply that the solution to (Pn∗) solves (Pn∗) as well.

Lemma 5 Consider a solution An to (Pn) and the corresponding Γn solving(17) defining V H(An; n). If the cutoff values T given by Lemma 2 at thissolution are given by T = {n− 1, N}, then An solves (Pn).

49

Proof. When T = {n − 1, N}, Lemma 2 implies that Γn also satisfies thefirst order conditions associated with the program defining V H(An; n), andtherefore solves that program. An is therefore feasible in (Pn). (Pn) is atighter program than (Pn), so An solves (Pn).

Lemma 6 Assume n∗ ≤ n ≤ n∗max. Let A′n = (a′n0 , · · · , a′nN) and Γ′n =(c′n0 , · · · , c′nN) solve (Pn) and the program defining V H(A′n; n), respectively,and let T be the associated cutoffs from Lemma 2. Then T = {n− 1, N}.

Proof. If T 6= {n − 1, N}, take the largest tk ∈ T less than N . For A′n

and Γ′n to solve (Pn) with cutoffs T and the program defining V H(A′n; n),respectively, Lemma 2 requires:

a′ntk ≤ c′ntkand

a′ntk+1 ≥ c′ntk+1.(27)

First suppose, by way of contradiction, that tk ≥ n∗max, where n∗max isdefined in Algorithm 1. Then combining (27) with the necessary condition(23), we observe:

(c′ntk)−γ

(1− ν

SHtk

SLtk

)≤ (c′ntk+1)

−γ

(1− ν

SHtk+1

SLtk+1

). (28)

Lemma 2 also requires:

SHtk

(c′ntk)−γ > SH

tk+1(c′ntk+1)

−γ. (29)

Combining (28) and (29) yields:

SHtk+1

SHtk

(1− ν

SHtk

SLtk

)<

(1− ν

SHtk+1

SLtk+1

)or ν <

1SH

tk+1− 1

SHtk

1SL

tk+1− 1

SLtk

.

This contradicts Lemma 1 when tk ≥ n∗max by Lemma 1.When T = {n− 1, N} at the solution to (Pn), the solutions to (Pn) and

(Pn) coincide by Lemma 5. Having ruled out tk ≥ n∗max, we conclude that(Pn∗max

) is uniquely solved with cutoffs Tn∗max= {n∗max − 1, N} and that the

solutions to (Pn∗max) and (Pn∗max

) coincide.

50

Proceeding by induction, suppose that for some n ≥ n∗, (Pn) is uniquelysolved with cutoffs Tn = {n − 1, N} for each n ≥ n + 1. By Lemma 5, thesolutions to (Pn) and (Pn) must then coincide for n ≥ n + 1. We will provethat Tn = {n − 1, N} by contradiction: suppose there is a solution to (Pn)involving cutoffs T = {n−1, · · · , tk, N} 6= {n−1, N}. From above, tk < n∗max

must hold.Fix µ = 1 (without loss of generality by Lemma 4), and take Γ′n and

A′n as in Lemma 4 for n = n and cutoffs T. Take Γ′′tk+1 and A′′tk+1 as inLemma 4 for n = tk + 1 and cutoffs {tk, N}; then Γ′′tk+1 = Γtk+1 = Γtk+1

and A′′tk+1 = Atk+1 = Atk+1 by the inductive hypothesis. By the argumentin the proof of Lemma 4, c′nt = c′′tk+1

t for t = tk + 1, · · · , N : having fixed µ,there is a unique solution within each interval, and the top intervals for thetwo problems coincide.

By Lemma 2, c′ntk ≥ a′ntk , whereby (23) yields µ ≡ 1 ≥ (a′ntk

)−γ(

1− νSH

tk

SLtk

).

Similarly, since a′′tk+1tk

= c′′tk+1tk

we conclude that 1 =(a′′tk+1

tk

)−γ(

1− νSH

tk

SLtk

).

Therefore, a′′tk+1tk

≤ a′ntk and c′′tk+1tk

≤ c′ntk .To complete the proof, note that if A′n solves (Pn) then Lemma 2 requires

SHtk

(c′ntk)−γ > SH

tk+1(c′ntk+1)

−γ. Since c′′tk+1tk

≤ c′ntk and c′′tk+1tk+1 = c′ntk+1, this implies

SHtk

(c′′tk+1tk

)−γ > SHtk+1(c

′′tk+1tk+1 )−γ. Noting that Γ′′tk+1 = Γtk+1 = Γtk+1, Algo-

rithm 1 implies n∗ ≥ tk + 1, since Algorithm 1 would have stopped at tk + 1,if not before. Since n ≥ n∗ and n ≤ tk, we have reached our contradiction,completing the proof.

We are now ready to state and prove Proposition 1. Proposition 1 statesthat the solution to (Pn∗) solves (P0). This means that (Pn∗) can be usedto solve (7) in the text—all that is additionally required is a search for theproper value of the multiplier ν. Since (Pn) is easily solved, we will be doneonce we have proved Proposition 1.

Proposition 1 If An∗ solves (Pn∗), then An∗ solves (P0) and (Pn∗) where n∗

is the outcome of Algorithm 1.

Proof. A solution An∗ = (an∗0 , · · · , an∗

N ) to (Pn∗) must exist, since theset of A satisfying the constraints is compact and the objective function iscontinuous. Lemmas 4 and 6 together imply that this solution is uniqueand involves the cutoff values T = {n − 1, N}. By Lemma 5, this solutionalso solves (Pn∗). Examination of the first order conditions shows that thissolution to (Pn∗) is unique.

51

Since V H(A; n) ≤ V H(A; 0) for every A, the value of Program (Pn∗)is at least as large as the value of Program (P0). It therefore suffices toshow that V H(An∗ ; n∗) = V H(An∗ ; 0). Let Γn∗ = (cn∗

0 , · · · , cn∗N ) solve the

program defining V H(An∗ ; n∗). Γn∗ must also solve the program (18) definingV H(An∗ ; n∗), or else An∗ couldn’t solve both (Pn) and (Pn). We need onlyto check that Γn∗ also solves the program (17) defining V H(An∗ , 0). Since(17) is a globally concave program and Γn∗ satisfies all of the constraints, it

suffices to show that SHt

(cn∗t

)−γ ≥ SHt+1

(cn∗t+1

)−γfor each t, with equality for

any t at which∑t

s=0 δs(cn∗s − an∗

s ) < 0.

For t ≥ n∗, SHt

(cn∗t

)−γ= SH

t+1

(cn∗t+1

)−γ. This is a necessary condition for

Γn∗ to solve the program defining V H(An∗ ; n∗). If n∗ = 0, we are done.Otherwise, for t < n∗, we have cn∗

t = an∗t , so

∑ts=0 δs(cn∗

s − an∗s ) = 0,

and we need only verify that SHt

(cn∗t

)−γ ≥ SHt+1

(cn∗t+1

)−γ. By Algorithm

1, SHn∗−1

(cn∗n∗−1

)−γ> SH

n∗(cn∗n∗

)−γ. We are therefore done if n∗ = 1.

If n∗ > 1, suppose, by way of contradiction, that

SHt

(cn∗t

)−γ< SH

t+1

(cn∗t+1

)−γ(30)

for some t < n∗ − 1. Since cn∗t = an∗

t for t < n∗,

(an∗

t

)−γ(

1− νSH

t

SLt

)=

(an∗

t+1

)−γ(

1− νSH

t+1

SLt+1

)(31)

by Lemma 3. (30) and (31) can be used to show that ν >

(1

SHt+1

− 1

SHt

1

SLt+1

− 1

SLt

). But

since t < n∗ ≤ n∗max, this is impossible given Algorithm 1 and Lemma 1.

This contradiction shows that SHt

(cn∗t

)−γ ≥ SHt+1 for each t < n∗ − 1, which

completes our proof.

52